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MIS 463. Analytic Hierarchy Process. The Analytic Hierarchy Process (AHP). It is popular and widely used method for multi-criteria decision making. Allows the use of qualitative, as well as quantitative criteria in evaluation. Founded by Saaty in 1980 .

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mis 463
MIS 463

Analytic Hierarchy Process

the analytic hierarchy process ahp
The Analytic Hierarchy Process (AHP)
  • It is popular and widely used method for multi-criteria decision making.
  • Allows the use of qualitative, as well as quantitative criteria in evaluation.
  • Founded by Saaty in 1980.
  • Wide range of applications exists:
    • Selecting a car for purchasing
    • Deciding upon a place to visit for vacation
    • Deciding upon an MBA program after graduation.
ahp general idea
AHP-General Idea
  • Develop an hierarchy of decision criteria and define the alternative courses of actions.
  • AHP algorithm is basically composed of two steps:

1. Determine the relative weights of the decision criteria

2. Determine the relative rankings (priority) of alternatives

! Both qualitative and quantitative information can be compared using informed judgements to derive weights and priorities.

example car selection
Example: Car Selection
  • Objective
    • Selecting a car
  • Criteria
    • Style, Reliability, Fuel-economy Cost?
  • Alternatives
    • Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata
hierarch y tree
Hierarchy tree

CivicSaturnEscortMiata

Alternative courses of action

ranking of criteria and alternatives
Ranking of Criteria and Alternatives
  • Pairwise comparisons are made with the grades ranging from 1-9.
  • A basic, but very reasonable, assumption:If attribute A is absolutely moreimportant than attribute B and is rated at 9, then B must be absolutely less importantthan A and is valued at 1/9.
  • These pairwise comparisons are carried out for all factors to be considered, usually notmore than 7, and the matrix is completed.
ranking of criteria

Style

Reliability

Fuel Economy

Style

1/1 1/2 3/1

2/1 1/1 4/1

Reliability

1/3 1/4 1/1

Fuel Economy

Ranking of criteria
ranking of priorities
Ranking of priorities
  • Consider [Ax = maxx] where
    • A is the comparison matrix of size n×n, for n criteria.
    • x is the Eigenvector of size n×1
    • maxis the Eigenvalue, max> n.
  • To find the ranking of priorities, namely the Eigen Vector X:

Initialization:

Take the squared power of matrix A, i.e., A2=A.A

Find the row sums of A2 and normalize this array to find E0.

Set A:=A2

Main:

1. Take the squared power of matrix A, i.e., A2=A.A

2. Find the row sums of A2 and normalize this array to find E1.

3. Find D= E1 - E0.

4. IF the elements of D are close to zero, then X= E1, STOP.

ELSE set A:=A2 , set E0:=E1 and go to Step 1.

slide10

1 0.5 3

2 1 4

0.33 0.25 1.0

0.3196

0.5584

0.1220

0.3194

0.5595

0.1211

  • Initialization:

3.00 1.75 8.00

5.33 3.0014.0

1.170.67 3.00

A=

A2=

Row sums

12.75

22.33

4.83

39.92

NormalizedRow Sums

0.3194

0.5595

0.1211

1.0

E0

  • Iteration 1:

Row sums

12.75

22.33

4.83

39.92

NormalizedRow Sums

0.3196

0.5584

0.1220

A2xA2=

E1

0.0002

-0.0011

0.0009

Almost zero, so

Eigen Vector, X = E1.

-

=

E1-E0 =

slide11
Criteria weights
  • Style .3196
  • Reliability .5584
  • Fuel Economy .1220
checking for consistency
Checking for Consistency
  • The next stage isto calculate a Consistency Ratio (CR) to measure how consistent the judgements havebeen relative to large samples of purely random judgements.
  • AHP evaluations are based on the aasumption that the decision maker is rational, i.e., if A is preferred to B and B is preferred to C, then A is preferred to C.
  • If the CR is greater than 0.1 the judgements are untrustworthy because they are too close for comfort torandomness and the exercise is valueless or must be repeated.
calculation of consistency ratio

1 0.5 3

2 1 4

0.333 0.25 1.0

0.3196

0.5584

0.1220

0.9648

1.6856

0.3680

0.3196

0.5584

0.1220

Calculation of Consistency Ratio
  • The next stage is to calculate max so as to lead to the Consistency Index and theConsistency Ratio.
  • Consider [Ax = max x] where x is the Eigenvector.

A x x

=max

=

λmax=average{0.9648/0.3196, 1.6856/0.5584, 0.3680/0.1220}=3.0180

  • Consistency index is found by
  • CI=(λmax-n)/(n-1)=(3.0180-3)/(3-1)= 0.009
consistency ratio
Consistency Ratio
  • The final step is to calculate the Consistency Ratio, CRby using the table below, derived from Saaty’s book,in which the upper row is the order of the random matrix, and the lower is thecorresponding index of consistency for random judgements.

Each of the numbers in this table is the average of CI’s derived from a sample of randomly selected reciprocal matrices using the AHP scale.

An inconsistency of 10% or less implies that the adjustment is small compared to the actual values of the eigenvector entries.

A CR as high as, say, 90% would mean that the pairwise judgements are just aboutrandom and are completely untrustworthy!

In the above example: CR=CI/0.58=0.0090/0.58=0.01552 (less than 0.1, so the evaluations are consistent)

ranking alternatives

.1160

.2470

.0600

.5770

.3790

.2900

.0740

.2570

Ranking alternatives

Eigenvector

Style

Civic

Saturn

Escort

Miata

Civic

1/1 1/4 4/1 1/6

Saturn

4/1 1/1 4/1 1/4

Escort

1/4 1/4 1/1 1/5

Miata

Miata

6/1 4/1 5/1 1/1

Reliability

Civic

Saturn

Escort

Miata

Civic

1/1 2/1 5/1 1/1

Saturn

1/2 1/1 3/1 2/1

Escort

1/5 1/3 1/1 1/4

Miata

1/1 1/2 4/1 1/1

slide16

Ranking alternatives

Normalized

Miles/gallon

Civic

34

.3010

Fuel Economy

Saturn

27

.2390

Escort

24

.2120

Miata

Miata

28

113

.2480

1.0

! Since fuel economy is a quantitative measure, fuel consumption ratios can be used to determine the relative ranking of alternatives; however this is not obligatory. Pairwise comparisons may still be used in some cases.

slide17

-Civic .1160

- Saturn .2470

- Escort .0600

- Miata .5770

-Civic .3790

- Saturn .2900

- Escort .0740

- Miata .2570

- Civic .3010

- Saturn .2390

- Escort .2120

- Miata .2480

ranking of alternatives

.3196

.5584

.1220

.2854

.2700

.0864

.3582

Civic

.1160 .3790 .3010

.2470 .2900 .2390

.0600 .0740 .2120

.5770 .2570 .2480

*

Saturn

=

Escort

Miata

Miata

Ranking of alternatives

Style Reliability Fuel

Economy

Criteria Weights

including cost as a decision criteria
Including Cost as a Decision Criteria

Adding “cost” as a a new criterion is very difficult in AHP. A new column and a new row will be added in the evaluation matrix. However, whole evaluation should be repeated since addition of a new criterion might affect the relative importance of other criteria as well!

Instead one may think of normalizing the costs directly and calculate the cost/benefit ratio for comparing alternatives!

  • CIVIC $12K .222 0.778
  • SATURN $15K .2778 1.028
  • ESCORT $9K .1667 1.929
  • MIATA $18K .333 0.930

Normalized

Cost

Cost/Benefits

Ratio

Cost

methods for including cost criterion
Methods for including cost criterion
  • Using graphical representations to make trade-offs.

cost

  • Calculate benefit/cost ratios
  • Use linear programming
  • Use seperate benefit and cost trees and then combine the results

benefit

complex decisions
Complex decisions
  • Many levels of criteria and sub-criteria exists for complex problems.
ahp software
AHP Software:

Professional commercial software Expert Choice developed by Expert Choice Inc. is available which simplifies the implementation of the AHP’s steps and automates many of its computations

  • computations
  • sensitivity analysis
  • graphs, tables
ex 2 evaluation of job offers

11/5 1/3 1/2

5 1 2 4

3 1/2 1 3

2 1/2 1/3 1

Location

Salary

Content

Long-term

Ex 2: Evaluation of Job Offers

Ex: Peter is offered 4 jobs from Acme Manufacturing (A), Bankers Bank (B), Creative Consulting (C), andDynamic Decision Making (D).

He bases his evaluation on the criteria such as location, salary, job content, andlong-term prospects.

Step 1: Decide upon the relative importance of the selection criteria:

Location Salary Content Long-term

a different way of calculating priority vectors

0.0910.102 0.091 0.059

0.455 0.513 0.545 0.471

0.273 0.256 0.273 0.353

0.182 0.128 0.091 0.118

Location

Salary

Content

Long-term

A Different Way of Calculating Priority Vectors:

1) Normalize the column entries by dividing each entry by the sum of the column.

2) Take the overall row averages

Location Salary Content Long-term Average

0.086

0.496

0.289

0.130

+ +

1 1 1 1 1

example 2 evaluation of job offers

A

B

C

D

A

B

C

D

0.161 0.137 0.171 0.227

0.322 0.275 0.257 0.312

0.484 0.549 0.514 0.409

0.032 0.040 0.057 0.045

1 1/2 1/3 5

2 1 1/2 7

3 2 1 9

1/5 1/7 1/9 1

Example 2: Evaluation of Job Offers

Location Scores

Step 2: Evaluate alternatives w.r.t. each criteria

A B C D

Relative Location Scores

A B C D Avg.

0.174

0.293

0.489

0.044

example 2 calculation of relative scores

A

B

C

D

0.174 0.050 0.210 0.510

0.293 0.444 0.038 0.012

0.489 0.312 0.354 0.290

0.044 0.194 0.398 0.188

Example 2: Calculation of Relative Scores

Relative weights for each criteria

Relative scores for each alternative

Relative Scores for Each Criteria

Location Salary Content Long-Term

0.086 0.496 0.289 0.130

0.164 0.256 0.335 0.238

x

=

more about ahp pros and cons
More about AHP: Pros and Cons
  • AHP is technique for formalizing decision making such that
  • It is applicable when it is difficult to formulate criteria evaluations, i.e., it allows qualitative evaluation as well as quantitative evaluation.
  • It is applicable for group decision making environments
  • However
  • There are hidden assumptions like consistency
  • Difficult to use when there are large number of evaluations Use GDSS
  • Use constraints to eliminate some alternatives
  • Difficult to add a new criterion or alternative Use cost/benefit ratio if applicable
  • Difficult to take out an existing criterion or alternative, sincethe best alternative might differ if the worst one is excluded.
group decision making
Group Decision Making
  • The AHP allows group decision making, where groupmembers can use their experience, values and knowledge to break down a problem into a hierarchy andsolve. Doing so provides:
    • Understand the conflicting ideas in the organization and try to reach a consensus.
    • Minimize dominance by a strong member of the group.
    • Members of the group may vote for the criteria to form the AHP tree. (Overall priorities are determined by the weighted averages of the priorities obtained from members of the group.)
  • However;
  • The GDSS does not replace all the requirements for group decision making. Open meetings with the involvement of all members are still an asset.
example 3 ahp in project management
Example 3: AHP in project management

Prequalification of contractors aims at the elimination of incompetent contractors from the bidding process.

It is the choice of the decision maker to eliminate contractor E from the AHP evalution since it is not “feasible” at all !!

example 3 ahp in project management30
Example 3: AHP in project management

Step 1: Evaluation of the weights of the criteria

Step 2: a) Pairwise comparison matrix for experience

example 3 ahp in project management31
Example 3: AHP in project management

Calculation of priority vector:

x

=

Probably Contractor-E should have been eliminated. It appears to be the worst.

Note that a DSS supports the decision maker, it can not replace him/her. Thus,

an AHP Based DSS should allow the decision maker to make sensitivity analysis of his judgements on the overall priorities !

references
References

Al Harbi K.M.A.S. (1999), Application of AHP in Project Management, International Journal of Project Management, 19, 19-27.

Haas R., Meixner, O., (2009) An Illustrated Guide to the Analytic Hierarchy Process, Lecture Notes, Institute of Marketing & Innovation, University of Natural Resources and http://www.boku.ac.at/mi/

Saaty, T.L., Vargas, L.G., (2001), Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, Kluwer’s Academic Publishers, Boston, USA.